Quantum Calabi-Yau and Classical Crystals

By A. Okounkov, N. Reshetikhin, C. Vafa. 2003

454 citations recorded by the source corpus.

We propose a new duality involving topological strings in the limit of the large string coupling constant. The dual is described in terms of a classical statistical mechanical model of crystal melting, where the temperature is the inverse of the string coupling constant. The crystal is a discretization of the toric base of the Calabi-Yau with lattice length g s. As a strong piece of evidence for this duality we recover the topological vertex in terms of the statistical mechanical probability distribution for crystal melting. We also propose a more general duality involving the dimer problem on periodic lattices and topological A-model string on arbitrary local toric threefolds. The (p, q) 5-brane web, dual to Calabi-Yau, gets identified with the transition regions of rigid dimer configurations.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/d75712f4aa21c8bb173933c55ace962fd1eb487f

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Citations in the corpus

Supersymmetric Gauge Theories, Intersecting Branes and Free Fermions (2007, background): “Let us recall that the latter is obtained by considering the (commuting) actions of the unitary group and symmetric group U(N) × Sk ⊂ U(Nk) on the vector space CNk, regarded as the k-th tensor product of the fundamental representation CN .”
The library of Babel: on the origin of gravitational thermodynamics (2005, background): “7. What is the relation between our analysis of typical half-BPS states and the description of quantum foam in the topological string [ 57 ]? In particular, there are strong similarities in the appearance of limit shapes of Young diagrams and topologically complex geometries containing random insert”
A mathematical theory of the topological vertex (2004, background): “This correspondence can also be formulated for certain non-compact threefolds in the presence of a torus action; the correspondence for toric Calabi-Yau threefolds is equivalent to the validity of the topological vertex [36, 42].”
Counting BPS operators in gauge theories: quivers, syzygies and plethystics (2006, background): “Therefore, the geometry is a cone over a torus! The generating function g(t) is the well-known MacMahon function [65].”
Limit shapes and the complex Burgers equation (2005, unknown): “There exist intriguing connections, first noticed in [19], between the random surfaces studied in this paper and topological sting theory, see e.”
Supergravity microstates for BPS black holes and black rings (2005, background): “’s geometric transitions picture [32, 33] and melting crystal space-time foam picture [35, 36] which will certainly attract future attention.”
Gromov–Witten theory and Donaldson–Thomas theory, I (2003, methodology): “g the equivariant localizationformula forDonaldsonThomas theory with the topological vertex expansion, speciﬁcally with the melting crystal interpretation of the topological vertex expansion found in [18, 34], we obtain our main result. 27 Theorem 3. For the toric local Calabi-Yau geometry, the equa”
Dimer models from mirror symmetry and quivering amoebae (2005, methodology): “derstood the role of dimer models in describing the tree-level superpotential of the quiver theories, can we also use the dimer models as a basis for studying quantum corrections? It is already known [16] (see also [46, 47]) that these same dimer models may be used to compute the topological string ”
Refined, Motivic, and Quantum (2009, methodology): “Of course, the quantity w∞(Θ ) must be the Donaldson-Thomas partition function of the conifold, and this relation holds because pyramid partitions of length n→∞ effectively reduce to the topological vertex formalism of [12,13] (see also [27,28]).”
A& B model approaches to surface operators and Toda thoeries (2010, methodology): “There exists an alternative way to obtain topological string partition functions using a relation to statistical models of crystal melting [49].”
Non-commutative Donaldson-Thomas theory and the conifold (2007, background): “The series ZA,1(q) can be computed by torus localization [22, 21, 4], the torus-fixed points being parametrized by 3-dimensional partitions α ⊂ N3.”
From Sasaki-Einstein spaces to quivers via BPS geodesics: Lp,q|r (2005, background): “The quiver/dimer ↔ toric Calabi-Yau’s correspondence is part of a general framework [31] connecting statistical mechanics of crystals and topological strings.”
Dimer models and toric diagrams (2005, background): “ A. Toric phases of Y6,0 21 1. Introduction Dimer models1 are two-dimensional statistical mechanical models of tilings of graphs. They have been recently introduced to the string theory literature in [1], which proposed to relate the dimer models to the target space topological string ﬁeld theory. T”